New Proof Shows the Maxwell Conjecture Is False for Point Charges
The Maxwell Conjecture Is False (GPT 5.6 Sol)
We present a specific arrangement of five point charges in Euclidean space that generates at least 24 non-degenerate critical points. This discovery directly contradicts Maxwell's conjecture, which claimed the maximum number of such points for n charges is (n-1)^2. Our findings demonstrate that the established limit for electrostatic potential critical points is incorrect.
Maxwell's conjecture that the field of n point charges has at most (n-1)^2 critical points which are all non-degenerate is therefore false.
- mellosouls
Not to denigrate the moment (AI ingress into theory which this is a part of) or the result here, but these headlines are perhaps overstating the importance - some of the theories and conjectures are available for AI-assisted exploration because they are quite niche and not very important.
Maxwell's name being invoked here for instance implies a hundred year old foundational problem like Fermat, but it's just a recent conjecture that was inspired by reflections from the great man on his work.
- d_burfoot
Tip for smart science-y young people: think about a career in experimental physics. Experimental data is the complement of theoretical power. Since theory can be provided cheaply by LLMs, experimental ability is now the bottleneck for progress in physics.
I expect to see frontier labs or startups hiring experimentalists to provide data for LLMs to analyze, pushing towards breakthroughs in areas like room-temperature superconductors and fusion.
- Syzygies
It is mathematical folklore that one should attempt to prove a conjecture by day, disprove it by night. Jordan Ellenberg recently popularized this in his 2014 book. He and I both heard this from Barry Mazur, but it dates at least to Bing, if not antiquity.
What is the purpose of mathematics? To be the architect of new conventions by seeing clearly past the old? If so, believing that the entire point is proving statements is a poor start. Bill Thurston was a visionary who happened to prove a great deal of what he saw, but his influence was his vision.
For those of us who like to understand every line of code we generate, and have labored for years to learn how to make best use of AI, a factor of two is a reasonable estimate for our productivity gain.
For those of us who believe mathematics is about achieving human understanding, having machines decide what's true and what isn't makes a night and day difference. Again, about a factor of two.
- beernet
On the one-hand side, it's really impressive how LLMs drive mathematics forward, and this pace is only accelerating very quickly.
At the same time, most of the proofs I've looked at appear super messy and chaotic to me (while still being correct of course, so it doesn't matter). LLMs do not care about "elegance" the way human beings do, which is a big advantage. LLMs for mathematics is such a great fit on many levels. Can't wait for a significant breakthrough, prove P=NP and all hell breaks loose.
- captainbland
This one is interesting as it's been hand verified. There was a recent proof that inadvertantly "proved" the collatz conjecture by triggering a bug in LEAN: https://infosec.exchange/@0xabad1dea/117002106099986943
- j_maffe
Visualization of the configuration: https://claude.ai/public/artifacts/9db65255-16ff-4f8e-8be1-1...
- JPLeRouzic
Please, what does that mean for Maxwell equations? For electromagnetism?
(Wikipedia redirects Maxwell's conjecture to Maxwell equations).
- vatsachak
Awesome! Confirms what we know; LLMs are superhuman at short term reasoning and breadth